[Paper Review] An Embedding theorem for abelian monoidal categories

Phùng Hô Hái|ArXiv.org|Apr 25, 2000

Algebraic structures and combinatorial models4 references3 citations

TL;DR

This paper establishes an embedding theorem for small abelian rigid monoidal categories into the category of bimodules over a ring, using a construction that extends the category to a Grothendieck monoidal category and then embeds it into a module category before realizing the bimodule structure. The key result is that such categories admit an exact monoidal embedding into R-Mod-R, with applications to semisimple symmetric categories showing they are Tannakian when the characteristic of the base field is zero.

ABSTRACT

We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.

Motivation & Objective

• To generalize the Freyd-Mitchell embedding theorem to the setting of abelian monoidal categories.
• To address the lack of a natural target category for monoidal embeddings in the braided or non-symmetric case.
• To construct a monoidal embedding of small abelian rigid monoidal categories into the category of bimodules over a ring.
• To show that semisimple symmetric monoidal categories with simple unit are Tannakian under characteristic zero.
• To provide an explicit realization of the embedding via infinite matrices of finite rank bimodules.

Proposed method

• Extend a small abelian monoidal category to a cocomplete Grothendieck monoidal category with an injective cogenerator.
• Construct a ring R such that the category embeds fully and faithfully into the category of R-bimodules.
• Use the tensor product structure to define a monoidal functor from a module category to the bimodule category.
• Represent objects as infinite matrices with finite row and column support, where the tensor product corresponds to matrix multiplication.
• Leverage rigidity to ensure images are finitely generated projective bimodules, enabling finite-dimensional estimates.
• Apply the coherence theorem and adjoint functor criteria to ensure compatibility with colimits and closed structures.

Experimental results

Research questions

• RQ1 Can a small abelian rigid monoidal category be embedded monoidally into the category of bimodules over a ring?
• RQ2 What conditions ensure that such an embedding is exact and faithful?
• RQ3 How does the structure of semisimple rigid monoidal categories behave under this embedding?
• RQ4 Can the embedding be used to prove Tannakian duality in the symmetric case?
• RQ5 What constraints does the bimodule representation impose on the growth of endomorphism spaces?

Key findings

• Any small abelian rigid monoidal category admits an exact monoidal embedding into the category of bimodules over a ring.
• The image of each object under the embedding corresponds to an infinite matrix with only finitely many non-zero entries per row and column.
• The tensor product of objects corresponds to matrix multiplication in the bimodule representation.
• For any object X in a semisimple rigid monoidal category with simple unit, the dimension of End(X^n) over the endomorphism ring K is bounded by d^n for some d depending on the category.
• If the category is symmetric and char(K) = 0, then for some n, the n-th antisymmetric power of any object X vanishes, implying the category is Tannakian.
• The categorical dimension of any object becomes an integer under a modified symmetry, which allows application of Deligne’s Tannakian theory.